HF-拓扑与HF-空间

HF-topology and HF-space

在构造划分拓扑的基础上引入了一类颇有价值的特殊的拓扑空间——划分拓扑空间。指出并论证了划分拓扑空间中开集与闭集的同一性;划分基与极小非空开集的关连性;在引入极小非空开集概念的基础上通过几个引理的建立,又论证了一系列颇有价值的理论:证明了在划分空间范围内HF-空间是正则空间、正规空间、局部连通空间的必然性;离散空间与T4空间的统一性,Lindeloff空间、A2空间、可分空间的三位一体性;Tychonoff空间、Hausdorff空间与T1空间的三位一体性;终于得到在划分空间范围内,T1空间、T2空间、T3空间、T3.5空间、T4空间五体合一这么一个结果。

A special topological space,namely the HF-space,was introduced in this paper, based on structuring HF-topology. It pointed out and demonstrated the identity of open set and closed set, and the relevance of HF-space and the minimum non-empty open set as well. Moreover, starting from the concept of the minimum non-empty open set,it also proved a series of significant theory, such as the necessity of the regularity of HF-space,normal space and locally connected space within HF-space;the unity of discrete space and T4- space ; the trinity of Lindeloff space, A2-space and separable space ; the trinity of Tychonoff space, Hausdorff space and T1-space. Perfect results among Tl-space T2-space T3-space T3.s-space and T4-space within HF- space were derived in the end.